A maximal T-space of F₃[x]₀

Abstract

In earlier work, we have established that for any finite field k, the free associative k-algebra on one generator x, denoted by k[x]₀, has infinitely many maximal T-spaces, but exactly two maximal T-ideals (each of which is a maximal T-space). However, aside from these two T-ideals, no specific examples of maximal T-spaces of k[x]₀ were determined at that time. In a subsequent work, we proposed that for a finite field k of characteristic p > 2 and order q, for each positive integer n which is a power of 2, the T-space Wn, generated by {x + xqⁿ, xqⁿ⁺¹}, is maximal, and we proved that W₁ is maximal. In this note, we prove that for q = p = 3, W₂ is maximal.